Heavenly Mathematics: The Mathematics of the Chinese, Indian

Heavenly Mathematics:
The Mathematics of the
Chinese, Indian, Islamic and
Gregorian Calendars
Helmer Aslaksen
Department of Mathematics
National University of Singapore
aslaksen@math.nus.edu.sg
www.math.nus.edu.sg/aslaksen/
www.chinesecalendar.net
1
Public Holidays
There are 11 public holidays in Singapore.
Three of them are secular.
1. New Year’s Day
2. Labour Day
3. National Day
The remaining eight cultural, racial or religious holidays consist of two Chinese, two
Muslim, two Indian and two Christian.
2
Cultural, Racial or Religious Holidays
1. Chinese New Year and day after
2. Good Friday
3. Vesak Day
4. Deepavali
5. Christmas Day
6. Hari Raya Puasa
7. Hari Raya Haji
Listed in order, except for the Muslim holidays, which can occur anytime during the
year.
Christmas Day falls on a fixed date, but all
the others move.
3
A Quick Course in Astronomy
The Earth revolves counterclockwise around
the Sun in an elliptical orbit. The Earth rotates counterclockwise around an axis that is
tilted 23.5 degrees.
March equinox
June
solstice
December
solstice
September equinox
N
E
W
June equi
sol
S
Dec
sol
Beijing
N
E
W
June equi
sol
S
Dec
sol
Singapore
In the northern hemisphere, the day will be
longest at the June solstice and shortest at
the December solstice. At the two equinoxes
day and night will be equally long. The equinoxes and solstices are called the seasonal
markers.
4
The Year
The tropical year (or solar year) is the time
from one March equinox to the next. The
mean value is 365.2422 days.
The synodic month is the time from one new
Moon to the next. It ranges from 29.27 days
to 29.84 days with a mean of 29.53 days.
12 × 29.5 = 354, so a lunar year consisting of
12 lunar months is about 11 days short of a
solar year.
365 − 12 × 29.5 = 11
365/11 ≈ 33
5
The Metonic Cycle
19 solar years is almost exactly 235 lunar
months.
235 × 29.53 = 6939.6884,
19 × 365.2422 = 6939.6018.
The difference is about two hours. This is
called the Metonic cycle (432 BCE). It was
known in China by about 600 BCE and was
called the zhāng (d) cycle.
The Metonic cycle is used in the Jewish calendar, in the computation of Easter, and was
used in the Chinese calendar before 104 BCE.
6
Classification of Calendars
solar Gregorian calendar. Basic unit is day.
Approximates the tropical year by adding
leap days. Ignores the Moon. The year
is 365 or 366 days.
lunar Islamic calendar. Basic unit is lunar
month. Ignores the Sun. The year is 12
months or 354 (sometimes 353 or 355)
days.
lunisolar Chinese and Jewish calendars. Basic unit is lunar month. Approximates the
tropical year by adding leap months. The
year is 12 or 13 months. A 12-month year
is 354 (sometimes 353 or 355) days. A
13-month year is 384 (sometimes 383 or
385) days.
The Chinese calendar is NOT a lunar calendar!
7
Alternative Classification of Calendars
arithmetical Gregorian and Jewish calendars.
Based on arithmetical formulas. Prediction and conversion between different arithmetical calendars is simple.
astronomical Islamic, Indian and Chinese calendars. Based on astronomical data. Prediction and conversion is hard.
8
The Gregorian Calendar
A normal year consists of 365 days, but leap
years have 366 days. Year n is a leap year if
n is divisible by 4, but not by 100 or if n is
divisible by 400. 1900 is not a leap year, but
2000 is.
In the Julian calendar, every fourth year is a
leap year.
The average length of the Gregorian year is
365.2425. The difference between this and
the tropical year will cause an error of about
one day in 2,500 years.
9
Computation of Easter
The rule of thumb is that Easter Sunday is
the first Sunday after the first full Moon on
or after the day of the March equinox. It will
fall between March 22 and April 25.
The actual rule is that Easter Sunday in year
y falls on day d in month m where d and m
are computed as follows (all remainders from
division are dropped).
c
n
k
i
i
i
=
=
=
=
=
=
j
j
l
m
d
=
=
=
=
=
y/100
y − 19 ∗ by/19c
b(c − 17)/25c
c − c/4 − b(c − k)/3c + 19 ∗ n + 15
i − 30 ∗ bi/30c
i − bi/28c ∗ (1 − bi/28c ∗ b29/(i + 1)c
∗b(21 − n)/11c)
y + by/4c + i + 2 − c + bc/4c
j − 7 ∗ bj/7c
i−j
3 + b(l + 40)/44c
l + 28 − 31 ∗ bm/4c
10
The Islamic Calendar
No leap months. Muslim holiday move about
11 days backward each year.
New months are determined by sighting of
the new Moon. A new Moon is normally
not visible until it is more than 24 hours old.
Muslim months start between one and four
days after the Chinese months (usually on
the third day). The Muslim month starts at
sunset the previous day.
Hari Raya Puasa is the first day of the 10th
month. Hari Raya Haji is the 10th day of the
12th month.
In 1999, Hari Raya Puasa fell on Jan. 19. In
2000, it will fall on Jan. 8 and on Dec. 27.
Since 365/11 ≈ 33, we see that such double
Hari Raya Puasa’s will occur every 32 or 33
years.
11
The Chinese Calendar
The goal is to approximate the solar year by
adding leap months. Sine 12 lunar months
are 11 days too short we will need to add a
leap month a little bit more than every third
year. In ancient times, this was done by observing nature.
Since 235 = 19 × 12 + 7, we can use the
Metonic cycle and get a decent lunisolar calendar by having 7 leap years in every 19-year
cycle.
12
Chinese New Year
It can be shown that Chinese New Year will
always fall between Jan. 21 and Feb. 21.
Most of the time Chinese New Year will fall
11 (or sometimes 10 or 12) days earlier than
the previous year, but if that would take us
outside of the Chinese New Year range of
Jan. 21 to Feb. 21, we must add a leap
month, so Chinese New Year jumps 19 (or
sometimes 18 or 20) days later.
1998 1999 2000 2001 2002 2003
28/1 16/2
5/2
24/1 12/2
1/2
19
11
12
19
11
2004 2005 2006 2007 2008 2009
22/1
9/2
29/1 18/2
7/2
26/1
10
18
11
20
11
12
13
Chinese New Year and Hari Raya Puasa
The Muslim holidays will always fall about
11 days earlier. Sometimes Hari Raya Puasa
will coincide with Chinese New Year. They
will then stay together for two or three years.
After that, HRP will continue on its 33 year
cycle backwards through the calendar, while
CNY will jump forward because of a leap
month. They will then meet up again in 30
or 31 years. This happened from 1964 to
1966, from 1996 to 1998, and will happen
again from 2029 to 2031.
14
The 19-year Cycle
Because of the Metonic cycle, there is almost
a 19-year cycle in the Chinese calendar. I
was born on April 16, 1960. This was the
21st day in the 3rd month in the Chinese
calendar. Normally my birthday will fall on
different days in the Chinese calendar, but
my 19th birthday fell on the 20th day in the
third month. The same goes for my 38th
and 57th birthday. So we see that the 19year cycle is close but not exact.
There are two reasons for this. First of all,
the Metonic cycle is off by about two hours.
But more importantly, we are now comparing
the Chinese calendar not with the tropical
year, but with the Gregorian calendar, which
is just an approximation to the tropical year.
In particular, since 19 is not a multiple of 4,
different cycles will contain different numbers
of leap years.
15
The 19-year Cycle and the Dates of
Chinese New Year
1980:
1981:
1982:
1983:
1984:
1985:
1986:
1987:
1988:
1989:
1990:
1991:
1992:
1993:
1994:
1995:
1996:
1997:
1998:
Feb.
Feb.
Jan.
Feb.
Feb.
Feb.
Feb.
Jan.
Feb.
Feb.
Jan.
Feb.
Feb.
Jan.
Feb.
Jan.
Feb.
Feb.
Jan.
16
5
25
13
2
20
9
29
17
6
27
15
4
23
10
31
19
7
28
1999:
2000:
2001:
2002:
2003:
2004:
2005:
2006:
2007:
2008:
2009:
2010:
2011:
2012:
2013:
2014:
2015:
2016:
2017:
Feb.
Feb.
Jan.
Feb.
Feb.
Jan.
Feb.
Jan.
Feb.
Feb.
Jan.
Feb.
Feb.
Jan.
Feb.
Jan.
Feb.
Feb.
Jan.
16
5
24
12
1
22
9
29
18
7
26
14
3
23
10
31
19
8
28
16
Dates of Chinese New Year Between
1645 and 2644
38
28
42
39
38
38
37
36
35
34 35 36
33 33
32 32 35 32
31
30
30 30
30
29 31
42
25
31 29
18
10
1
21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
17
The 24 Jié Qì
A fundamental concept in the Chinese calendar is the 24 solar terms or jié qì (dd).
They are a generalization of the solstices and
equinoxes. The even ones are called major
solar terms or zhōng qì (dd).
A useful rule of thumb is that Chinese New
Year is the new Moon closest to lì chūn (d
d), the beginning of spring. This rule is correct most of the time, but it failed in 1985
and will fail again in 2015. Since lì chūn falls
around Feb. 4, this helps explain why Chinese New Year will always fall between Jan.
21 and Feb. 21. It also helps explain why
Chinese New Year is called the spring festival.
In Western astronomy, spring begins at spring
equinox. In Chinese astronomy, spring begins
midway between winter solstice and spring
equinox.
18
The 24 Jié Qì
J1
Z1
J2
Z2
J3
Z3
J4
Z4
J5
Z5
J6
Z6
J7
Z7
J8
Z8
J9
Z9
J10
Z10
J11
Z11
J12
Z12
Lì chūn
Yǔ shuı̌
Jı̄ng zhé
Chūn fēn
Qı̄ng míng
Gǔ yǔ
Lì xià
Xiǎo mǎn
Máng zhòng
Xià zhì
Xiǎo shǔ
Dà shǔ
Lì qiū
Chǔ shǔ
Bái lù
Qiū fēn
Hán lù
Shuāng jiàng
Lì dōng
Xiǎo xuě
Dà xuě
Dōng zhì
Xiǎo hán
Dà hán
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
Beginning of spring
Rain water
Waking of insects
Spring equinox
Pure brightness
Grain rain
Beginning of summer
Grain full
Grain in ear
Summer solstice
Slight heat
Great heat
Beginning of autumn
Limit of heat
White dew
Autumnal equinox
Cold dew
Descent of frost
Beginning of winter
Slight snow
Great snow
Winter solstice
Slight cold
Great cold
February 4
February 19
March 6
March 21
April 5
April 20
May 6
May 21
June 6
June 22
July 7
July 23
August 8
August 23
September 8
September 23
October 8
October 24
November 8
November 22
December 7
December 22
January 6
January 20
19
The Chinese Meridian
Calculations are based on the meridian 120◦
East.
Before 1929 the computations were based on
the meridian in Beijing (116◦250), but in 1928
China adopted a standard time zone based on
120◦ East.
20
The Length of a Chinese Month
The day on which a new Moon occurs is the
first day of the new month.
The length of the months are determined astronomically. The lunar month can vary between about 29.25 and 29.75 with a mean of
29.53. Suppose a month is 29.5 days.
New Moon
May 1 13h
May 1 1h
Next new Moon
May 31 1h
May 30 13h
Length
30 days
29 days
There can be four long months or three short
months in a row.
21
The Mid-Autumn Festival
If the 1st month marks the beginning of spring,
autumn should start with the 7th month.
This explains why the Mid-Autumn Festival is
celebrated on the 15th day of the 8th month.
22
The Chinese Solar Calendar
It is important to understand that the Chinese calendar is a combination of two calendars, the usual lunisolar calendar and a solar
calendar that follows the 24 jié qì. The solar calendar is traditionally called the farmer’s
calendar (dd). Unfortunately the term farmer’s calendar has come to include the lunisolar calendar. The Chinese solar calendar follows the tropical year closely, so it is perfect
for farming purposes, but the lunisolar calendar is not at all suitable for farmers.
There are two Chinese holidays that are determined by the solar calendar, namely qı̄ng
míng (dd) around Apr. 5 and winter solstice
dōng zhì (dd) around Dec. 22.
Notice that lunar dates can fall within a range
of about one month in the solar calendar and
conversely. Chinese New Year can fall between Jan. 21 and Feb. 21, while qı̄ng míng
can fall between the 13th day of the 2nd
month and the 17th day of the 3rd month.
23
The Chinese Year
There are several years in the Chinese calendar. The most important are the suì (d)
and the nián (d).
A suì is the solstice year from one winter solstice to the next. This is the same as the
tropical year.
A nián is the Chinese year from one Chinese
New Year to the next. Since a Chinese year
can contain 12 or 13 lunar months, and they
can have 29 or 30 days, it can be shown that
the length of a nián can be 353, 354, 355,
383, 384 or 385 days long.
Just like we can think of the Gregorian year
as an approximation to the tropical year, we
can think of the nián as an approximation to
the suì.
The Chinese astrological year runs from lì
chūn, the beginning of spring about Feb. 5,
not from Chinese New Year.
24
Vesak Day
Traditionally, Buddhists have observed Vesak
Day on the 8th or 15th day of the fourth
month. Since the 1950’s the Singapore Buddhist Federation celebrates it on the first full
Moon in May.
25
Deepavali
Deepavali falls on the last day of the lunar month Asvina, called Purattasi in Tamil.
Like the Chinese calendar, the Tamil calendar consists of a solar calendar and a lunisolar
calendar. The solar month Asvina runs from
around Sep. 16 to around Oct. 17. The lunar
month Asvina starts with the first new Moon
within this solar month. It follows that Deepavali can fall between Oct. 15 and Nov. 15.
26
The Sexagenary Cycle
Heavenly Stems
1
2
3
4
5
6
7
8
9
10
dd
d
d
d
d
d
d
d
d
d
d
Earthly Branches
1
2
3
4
5
6
7
8
9
10
11
12
tiān gān
jiǎ
yı̌
bı̌ng
dı̄ng
wù
jı̌
gēng
xı̄n
rén
guı̌
dd
d
d
d
d
d
d
d
d
d
d
d
d
dì zhı̄
zı̌
chǒu
yín
mǎo
chén
sì
wǔ
wèi
shēn
yǒu
xū
hài
Element
Wood
Wood
Fire
Fire
Earth
Earth
Metal
Metal
Water
Water
Animal
Rat
Ox
Tiger
Rabbit
Dragon
Snake
Horse
Goat
Monkey
Chicken
Dog
Pig
27
The Golden Dragon
Let us denote both the stems and the branches by their numbers. We denote 1 by (1, 1)
or (d, d), 2 by (2, 2) or (d, d) and so
on up to (10, 10) or (d, d). But now we
have run out of stems, so we denote 11 by
(1, 11) or (d, d) and 12 by (2, 12) or (d,
d). Now we have run out of branches, too,
so 13 becomes (3, 1) or (d, d). We continue
in this way through 6 cycles of stems and 5
cycles of branches up to 60, which is (10, 12)
or (d, d). The next number is then (1, 1)
or (d, d), which starts a new sexagesimal
cycle.
Notice that each branch, or animal, occurs
five times in each 60-year cycle. An animal
corresponding to an odd number, will meet
the stems that correspond to the odd numbers. Year 2000 is the 17th year in the current cycle (see below), so it corresponds to
(7, 5) (17 = 10 + 7 = 12 + 5) or (d, d). So
we see that it is a metal dragon year, or a
“Golden Dragon".
28
The Eight Characters
This cycle is used for keeping track of years,
months, days and (double) hours in Chinese
astrology. Your date and time of birth is determined by the “Eight Characters" (d d)
formed by the pair of cyclical characters for
the year, month, day and hour. The 60-day
cycle has been used for keeping track of days
since ancient times. During the Hàn (d) dynasty, the 60-year cycle was also introduced.
29
What is Year 2000 in the Chinese
Calendar?
The Chinese do not have a continuous year
count. They started counting from one again
with each new emperor. However, from the
Hàn dynasty, some scholars tried to reconstruct the ancient Chinese chronology, and it
became customary to claim that the calendar
was invented by the Yellow Emperor, Huáng
Dì (dd), in 2637 BCE during the 61st year
of his reign. However, many people prefer to
start the count with the first year of his reign
in 2697 BCE. Since these years are 60 years
apart, it follows that 1984 was the first year
of either the 78th or 79th 60-year cycle. Using this as a starting point, Chinese New Year
in 2000 marks the beginning of the Chinese
year 4637 or 4697. While Chinese chronology
is fairly reliable going back to 841 BCE, and
oracle bones with date inscription go back
to the 13th century BCE, modern scholars
consider the Yellow Emperor to be a mythological figure.
30
Why Was the Calendar Important?
With a lunar or lunisolar calendar, errors are
much more obvious than with a solar calendar. A solar calendar can be off by a couple of weeks without anybody noticing. The
reason why the Catholic church had to reform the Julian calendar was because the
rules for computing Easter had frozen the
March equinox to be March 21. That meant
that Easter was drifting noticeably towards
summer. Otherwise, few would have cared
about the drift of the March equinox. But
with a lunar calendar, an error of even a
couple of days is a serious problem. Every
peasant could each month see that the new
Moon was visible near the end of the previous month or that the old Moon was visible
in the next month.
Because of the importance the Chinese rulers
placed on calendars, they were surprisingly
open to incorporate foreign ideas into the
making of calendars. The last three main calendar reforms have all been associated with
foreign impulses.
31
The Main Calendar Reforms
Before 621 BCE, the start of the month was
based on visibility of the crescent Moon. During the Zhōu (d) dynasty, the Metonic cycle
was used for determining leap months and
the leap months were always placed at the
end of the year. After the Tài Chū (dd)
calendar reform in 104 BCE, the “no zhōng
qì" (ddd) rule was used for determining
leap months, and the month containing the
December solstice was fixed to be the 11th
month.
The Táng (d) dynasty calendar reform in
619 switched to following the true Moon.
This was inspired by Indian Buddhist astronomers.
The Yuán (d) dynasty reform in 1280 was
inspired by Muslim astronomers. It was the
most accurate calendar in the world at that
time.
The last calendar reform came in 1645 during
the Qı̄ng dynasty (d) and was implemented
by Jesuit missionaries. It used the true Sun.
32
The Jesuits
In 1644, the German Adam Schall went to
the new Qı̄ng rulers and presented his calculations for an upcoming solar eclipse. He
challenged the Chinese and the Muslim astronomers in the Bureau, and the Jesuits’
calculations were best. Schall was appointed
director of the Bureau. The next year, he
formulated the current rules for the Chinese
calendar.
33
The Trial of the Jesuits
A Chinese official, Yáng Guāng Xiān (ddd),
had as his slogan that it was “better to have
a wrong calendar than to have foreigners in
China". Yáng managed to have the Jesuits
arrested in 1664. A solar eclipse was coming
up and while in prison, the Jesuits predicted
it would occur at 3 pm, Yáng predicted 2.15
pm, and the Muslim Wú Míng Xuǎn (ddd)
predicted 2.30 pm. On the day of the eclipse,
the Jesuits were brought into the palace in
chains, and everybody watched as the eclipse
occurred at 3pm sharp! Unfortunately, the
regents were not impressed and the Jesuits
were sentenced to death. However, the next
day a strong earthquake struck Beijing. This
was taken as a sign from Heaven that the
sentence was unjust, and the sentence of the
Jesuits was first converted to flogging and
eventually to just house arrest.
34
The Kang Xi Emperor
In 1668, the Kāng Xı̄ (dd) emperor took
over from the regents. The emperor ordered
the Belgian Verbiest, Yáng and Wú to compute the length of the shadow of a pole on
a certain day and the position of the Sun
at noon on a certain day. They were to
leave their instruments pointing towards the
predicted spot in the emperor’s garden two
weeks in advance. Verbiest easily won and
was appointed director of the Bureau, while
Yáng and Wú were arrested. Verbiest became personal tutor to the Kāng Xı̄ emperor,
and even learned Manchu. Jesuits remained
as directors of the Bureau until 1746 and it
was run by other Westerners until 1826.
35